3.13.81 \(\int \frac {(A+B x) \sqrt {d+e x}}{(a-c x^2)^2} \, dx\)

Optimal. Leaf size=225 \[ -\frac {\left (-\sqrt {a} A \sqrt {c} e-a B e+2 A c d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{5/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {\left (\sqrt {a} A \sqrt {c} e-a B e+2 A c d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{4 a^{3/2} c^{5/4} \sqrt {\sqrt {a} e+\sqrt {c} d}}+\frac {\sqrt {d+e x} (a B+A c x)}{2 a c \left (a-c x^2\right )} \]

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Rubi [A]  time = 0.40, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {821, 827, 1166, 208} \begin {gather*} -\frac {\left (-\sqrt {a} A \sqrt {c} e-a B e+2 A c d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{5/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {\left (\sqrt {a} A \sqrt {c} e-a B e+2 A c d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{4 a^{3/2} c^{5/4} \sqrt {\sqrt {a} e+\sqrt {c} d}}+\frac {\sqrt {d+e x} (a B+A c x)}{2 a c \left (a-c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*Sqrt[d + e*x])/(a - c*x^2)^2,x]

[Out]

((a*B + A*c*x)*Sqrt[d + e*x])/(2*a*c*(a - c*x^2)) - ((2*A*c*d - a*B*e - Sqrt[a]*A*Sqrt[c]*e)*ArcTanh[(c^(1/4)*
Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(4*a^(3/2)*c^(5/4)*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) + ((2*A*c*d - a*B
*e + Sqrt[a]*A*Sqrt[c]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a^(3/2)*c^(5/4)*Sqr
t[Sqrt[c]*d + Sqrt[a]*e])

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^m*
(a + c*x^2)^(p + 1)*(a*g - c*f*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*
x^2)^(p + 1)*Simp[a*e*g*m - c*d*f*(2*p + 3) - c*e*f*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x
] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 827

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^2} \, dx &=\frac {(a B+A c x) \sqrt {d+e x}}{2 a c \left (a-c x^2\right )}-\frac {\int \frac {\frac {1}{2} (-2 A c d+a B e)-\frac {1}{2} A c e x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{2 a c}\\ &=\frac {(a B+A c x) \sqrt {d+e x}}{2 a c \left (a-c x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} A c d e+\frac {1}{2} e (-2 A c d+a B e)-\frac {1}{2} A c e x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt {d+e x}\right )}{a c}\\ &=\frac {(a B+A c x) \sqrt {d+e x}}{2 a c \left (a-c x^2\right )}-\frac {\left (2 A c d-a B e-\sqrt {a} A \sqrt {c} e\right ) \operatorname {Subst}\left (\int \frac {1}{c d-\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 a^{3/2} \sqrt {c}}+\frac {\left (2 A c d-a B e+\sqrt {a} A \sqrt {c} e\right ) \operatorname {Subst}\left (\int \frac {1}{c d+\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 a^{3/2} \sqrt {c}}\\ &=\frac {(a B+A c x) \sqrt {d+e x}}{2 a c \left (a-c x^2\right )}-\frac {\left (2 A c d-a B e-\sqrt {a} A \sqrt {c} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{5/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {\left (2 A c d-a B e+\sqrt {a} A \sqrt {c} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{4 a^{3/2} c^{5/4} \sqrt {\sqrt {c} d+\sqrt {a} e}}\\ \end {align*}

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Mathematica [A]  time = 0.47, size = 375, normalized size = 1.67 \begin {gather*} \frac {-\frac {\sqrt [4]{c} \left (a A e^2+2 a B d e-3 A c d^2\right ) \left (\sqrt {\sqrt {c} d-\sqrt {a} e} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )-\sqrt {\sqrt {a} e+\sqrt {c} d} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )\right )}{2 \sqrt {a}}+\frac {c (d+e x)^{3/2} (-a A e+a B (d-e x)+A c d x)}{c x^2-a}+\frac {(a B e-A c d) \left (2 \sqrt {a} \sqrt [4]{c} e \sqrt {d+e x}+\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )-\left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )\right )}{2 \sqrt {a} \sqrt [4]{c}}}{2 a c \left (a e^2-c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*Sqrt[d + e*x])/(a - c*x^2)^2,x]

[Out]

((c*(d + e*x)^(3/2)*(-(a*A*e) + A*c*d*x + a*B*(d - e*x)))/(-a + c*x^2) - (c^(1/4)*(-3*A*c*d^2 + 2*a*B*d*e + a*
A*e^2)*(Sqrt[Sqrt[c]*d - Sqrt[a]*e]*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] - Sqrt[Sqrt[c
]*d + Sqrt[a]*e]*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]]))/(2*Sqrt[a]) + ((-(A*c*d) + a*B
*e)*(2*Sqrt[a]*c^(1/4)*e*Sqrt[d + e*x] + (Sqrt[c]*d - Sqrt[a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sq
rt[c]*d - Sqrt[a]*e]] - (Sqrt[c]*d + Sqrt[a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]
*e]]))/(2*Sqrt[a]*c^(1/4)))/(2*a*c*(-(c*d^2) + a*e^2))

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IntegrateAlgebraic [A]  time = 1.41, size = 337, normalized size = 1.50 \begin {gather*} -\frac {\sqrt {-\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right )} \left (\sqrt {a} A \sqrt {c} e-a B e+2 A c d\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {-\sqrt {a} \sqrt {c} e-c d}}{\sqrt {a} e+\sqrt {c} d}\right )}{4 a^{3/2} c^{3/2} \left (\sqrt {a} e+\sqrt {c} d\right )}-\frac {\sqrt {-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )} \left (-\sqrt {a} A \sqrt {c} e-a B e+2 A c d\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {a} \sqrt {c} e-c d}}{\sqrt {c} d-\sqrt {a} e}\right )}{4 a^{3/2} c^{3/2} \left (\sqrt {a} e-\sqrt {c} d\right )}+\frac {e \sqrt {d+e x} (a B e+A c (d+e x)-A c d)}{2 a c \left (a e^2-c d^2+2 c d (d+e x)-c (d+e x)^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*Sqrt[d + e*x])/(a - c*x^2)^2,x]

[Out]

(e*Sqrt[d + e*x]*(-(A*c*d) + a*B*e + A*c*(d + e*x)))/(2*a*c*(-(c*d^2) + a*e^2 + 2*c*d*(d + e*x) - c*(d + e*x)^
2)) - (Sqrt[-(Sqrt[c]*(Sqrt[c]*d + Sqrt[a]*e))]*(2*A*c*d - a*B*e + Sqrt[a]*A*Sqrt[c]*e)*ArcTan[(Sqrt[-(c*d) -
Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/(4*a^(3/2)*c^(3/2)*(Sqrt[c]*d + Sqrt[a]*e)) - (Sqr
t[-(Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e))]*(2*A*c*d - a*B*e - Sqrt[a]*A*Sqrt[c]*e)*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sq
rt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/(4*a^(3/2)*c^(3/2)*(-(Sqrt[c]*d) + Sqrt[a]*e))

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fricas [B]  time = 2.43, size = 3195, normalized size = 14.20

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+a)^2,x, algorithm="fricas")

[Out]

1/8*((a*c^2*x^2 - a^2*c)*sqrt((4*A^2*c^2*d^3 - 4*A*B*a*c*d^2*e + 2*A*B*a^2*e^3 + (B^2*a^2 - 3*A^2*a*c)*d*e^2 +
(a^3*c^3*d^2 - a^4*c^2*e^2)*sqrt((4*A^2*B^2*c^2*d^2*e^4 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^5 + (B^4*a^2 + 2*A^2*
B^2*a*c + A^4*c^2)*e^6)/(a^3*c^7*d^4 - 2*a^4*c^6*d^2*e^2 + a^5*c^5*e^4)))/(a^3*c^3*d^2 - a^4*c^2*e^2))*log((8*
A^3*B*c^3*d^3*e^2 - 4*(3*A^2*B^2*a*c^2 + A^4*c^3)*d^2*e^3 + 2*(3*A*B^3*a^2*c + A^3*B*a*c^2)*d*e^4 - (B^4*a^3 -
A^4*a*c^2)*e^5)*sqrt(e*x + d) + (2*A^2*B*a^2*c^3*d^2*e^3 - (3*A*B^2*a^3*c^2 + A^3*a^2*c^3)*d*e^4 + (B^3*a^4*c
+ A^2*B*a^3*c^2)*e^5 + (2*A*a^3*c^6*d^4 - B*a^4*c^5*d^3*e - 3*A*a^4*c^5*d^2*e^2 + B*a^5*c^4*d*e^3 + A*a^5*c^4
*e^4)*sqrt((4*A^2*B^2*c^2*d^2*e^4 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^5 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^6)
/(a^3*c^7*d^4 - 2*a^4*c^6*d^2*e^2 + a^5*c^5*e^4)))*sqrt((4*A^2*c^2*d^3 - 4*A*B*a*c*d^2*e + 2*A*B*a^2*e^3 + (B^
2*a^2 - 3*A^2*a*c)*d*e^2 + (a^3*c^3*d^2 - a^4*c^2*e^2)*sqrt((4*A^2*B^2*c^2*d^2*e^4 - 4*(A*B^3*a*c + A^3*B*c^2)
*d*e^5 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^6)/(a^3*c^7*d^4 - 2*a^4*c^6*d^2*e^2 + a^5*c^5*e^4)))/(a^3*c^3*d
^2 - a^4*c^2*e^2))) - (a*c^2*x^2 - a^2*c)*sqrt((4*A^2*c^2*d^3 - 4*A*B*a*c*d^2*e + 2*A*B*a^2*e^3 + (B^2*a^2 - 3
*A^2*a*c)*d*e^2 + (a^3*c^3*d^2 - a^4*c^2*e^2)*sqrt((4*A^2*B^2*c^2*d^2*e^4 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^5 +
(B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^6)/(a^3*c^7*d^4 - 2*a^4*c^6*d^2*e^2 + a^5*c^5*e^4)))/(a^3*c^3*d^2 - a^4*
c^2*e^2))*log((8*A^3*B*c^3*d^3*e^2 - 4*(3*A^2*B^2*a*c^2 + A^4*c^3)*d^2*e^3 + 2*(3*A*B^3*a^2*c + A^3*B*a*c^2)*d
*e^4 - (B^4*a^3 - A^4*a*c^2)*e^5)*sqrt(e*x + d) - (2*A^2*B*a^2*c^3*d^2*e^3 - (3*A*B^2*a^3*c^2 + A^3*a^2*c^3)*d
*e^4 + (B^3*a^4*c + A^2*B*a^3*c^2)*e^5 + (2*A*a^3*c^6*d^4 - B*a^4*c^5*d^3*e - 3*A*a^4*c^5*d^2*e^2 + B*a^5*c^4*
d*e^3 + A*a^5*c^4*e^4)*sqrt((4*A^2*B^2*c^2*d^2*e^4 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^5 + (B^4*a^2 + 2*A^2*B^2*a*
c + A^4*c^2)*e^6)/(a^3*c^7*d^4 - 2*a^4*c^6*d^2*e^2 + a^5*c^5*e^4)))*sqrt((4*A^2*c^2*d^3 - 4*A*B*a*c*d^2*e + 2*
A*B*a^2*e^3 + (B^2*a^2 - 3*A^2*a*c)*d*e^2 + (a^3*c^3*d^2 - a^4*c^2*e^2)*sqrt((4*A^2*B^2*c^2*d^2*e^4 - 4*(A*B^3
*a*c + A^3*B*c^2)*d*e^5 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^6)/(a^3*c^7*d^4 - 2*a^4*c^6*d^2*e^2 + a^5*c^5*
e^4)))/(a^3*c^3*d^2 - a^4*c^2*e^2))) + (a*c^2*x^2 - a^2*c)*sqrt((4*A^2*c^2*d^3 - 4*A*B*a*c*d^2*e + 2*A*B*a^2*e
^3 + (B^2*a^2 - 3*A^2*a*c)*d*e^2 - (a^3*c^3*d^2 - a^4*c^2*e^2)*sqrt((4*A^2*B^2*c^2*d^2*e^4 - 4*(A*B^3*a*c + A^
3*B*c^2)*d*e^5 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^6)/(a^3*c^7*d^4 - 2*a^4*c^6*d^2*e^2 + a^5*c^5*e^4)))/(a
^3*c^3*d^2 - a^4*c^2*e^2))*log((8*A^3*B*c^3*d^3*e^2 - 4*(3*A^2*B^2*a*c^2 + A^4*c^3)*d^2*e^3 + 2*(3*A*B^3*a^2*c
+ A^3*B*a*c^2)*d*e^4 - (B^4*a^3 - A^4*a*c^2)*e^5)*sqrt(e*x + d) + (2*A^2*B*a^2*c^3*d^2*e^3 - (3*A*B^2*a^3*c^2
+ A^3*a^2*c^3)*d*e^4 + (B^3*a^4*c + A^2*B*a^3*c^2)*e^5 - (2*A*a^3*c^6*d^4 - B*a^4*c^5*d^3*e - 3*A*a^4*c^5*d^2
*e^2 + B*a^5*c^4*d*e^3 + A*a^5*c^4*e^4)*sqrt((4*A^2*B^2*c^2*d^2*e^4 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^5 + (B^4*a
^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^6)/(a^3*c^7*d^4 - 2*a^4*c^6*d^2*e^2 + a^5*c^5*e^4)))*sqrt((4*A^2*c^2*d^3 - 4*A
*B*a*c*d^2*e + 2*A*B*a^2*e^3 + (B^2*a^2 - 3*A^2*a*c)*d*e^2 - (a^3*c^3*d^2 - a^4*c^2*e^2)*sqrt((4*A^2*B^2*c^2*d
^2*e^4 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^5 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^6)/(a^3*c^7*d^4 - 2*a^4*c^6*d
^2*e^2 + a^5*c^5*e^4)))/(a^3*c^3*d^2 - a^4*c^2*e^2))) - (a*c^2*x^2 - a^2*c)*sqrt((4*A^2*c^2*d^3 - 4*A*B*a*c*d^
2*e + 2*A*B*a^2*e^3 + (B^2*a^2 - 3*A^2*a*c)*d*e^2 - (a^3*c^3*d^2 - a^4*c^2*e^2)*sqrt((4*A^2*B^2*c^2*d^2*e^4 -
4*(A*B^3*a*c + A^3*B*c^2)*d*e^5 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^6)/(a^3*c^7*d^4 - 2*a^4*c^6*d^2*e^2 +
a^5*c^5*e^4)))/(a^3*c^3*d^2 - a^4*c^2*e^2))*log((8*A^3*B*c^3*d^3*e^2 - 4*(3*A^2*B^2*a*c^2 + A^4*c^3)*d^2*e^3 +
2*(3*A*B^3*a^2*c + A^3*B*a*c^2)*d*e^4 - (B^4*a^3 - A^4*a*c^2)*e^5)*sqrt(e*x + d) - (2*A^2*B*a^2*c^3*d^2*e^3 -
(3*A*B^2*a^3*c^2 + A^3*a^2*c^3)*d*e^4 + (B^3*a^4*c + A^2*B*a^3*c^2)*e^5 - (2*A*a^3*c^6*d^4 - B*a^4*c^5*d^3*e
- 3*A*a^4*c^5*d^2*e^2 + B*a^5*c^4*d*e^3 + A*a^5*c^4*e^4)*sqrt((4*A^2*B^2*c^2*d^2*e^4 - 4*(A*B^3*a*c + A^3*B*c^
2)*d*e^5 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^6)/(a^3*c^7*d^4 - 2*a^4*c^6*d^2*e^2 + a^5*c^5*e^4)))*sqrt((4*
A^2*c^2*d^3 - 4*A*B*a*c*d^2*e + 2*A*B*a^2*e^3 + (B^2*a^2 - 3*A^2*a*c)*d*e^2 - (a^3*c^3*d^2 - a^4*c^2*e^2)*sqrt
((4*A^2*B^2*c^2*d^2*e^4 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^5 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^6)/(a^3*c^7*
d^4 - 2*a^4*c^6*d^2*e^2 + a^5*c^5*e^4)))/(a^3*c^3*d^2 - a^4*c^2*e^2))) - 4*(A*c*x + B*a)*sqrt(e*x + d))/(a*c^2
*x^2 - a^2*c)

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giac [B]  time = 0.54, size = 436, normalized size = 1.94 \begin {gather*} \frac {{\left (2 \, A a c^{3} d^{2} - B a^{2} c^{2} d e - \sqrt {a c} A c d {\left | a \right |} {\left | c \right |} e - A a^{2} c^{2} e^{2} + \sqrt {a c} B a {\left | a \right |} {\left | c \right |} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c^{2} d + \sqrt {a^{2} c^{4} d^{2} - {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} a c^{2}}}{a c^{2}}}}\right )}{4 \, {\left (a^{2} c^{2} e - \sqrt {a c} a c^{2} d\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | a \right |}} + \frac {{\left (2 \, A a c^{3} d^{2} - B a^{2} c^{2} d e + \sqrt {a c} A c d {\left | a \right |} {\left | c \right |} e - A a^{2} c^{2} e^{2} - \sqrt {a c} B a {\left | a \right |} {\left | c \right |} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c^{2} d - \sqrt {a^{2} c^{4} d^{2} - {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} a c^{2}}}{a c^{2}}}}\right )}{4 \, {\left (a^{2} c^{2} e + \sqrt {a c} a c^{2} d\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | a \right |}} - \frac {{\left (x e + d\right )}^{\frac {3}{2}} A c e - \sqrt {x e + d} A c d e + \sqrt {x e + d} B a e^{2}}{2 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} - a e^{2}\right )} a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+a)^2,x, algorithm="giac")

[Out]

1/4*(2*A*a*c^3*d^2 - B*a^2*c^2*d*e - sqrt(a*c)*A*c*d*abs(a)*abs(c)*e - A*a^2*c^2*e^2 + sqrt(a*c)*B*a*abs(a)*ab
s(c)*e^2)*arctan(sqrt(x*e + d)/sqrt(-(a*c^2*d + sqrt(a^2*c^4*d^2 - (a*c^2*d^2 - a^2*c*e^2)*a*c^2))/(a*c^2)))/(
(a^2*c^2*e - sqrt(a*c)*a*c^2*d)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(a)) + 1/4*(2*A*a*c^3*d^2 - B*a^2*c^2*d*e + sq
rt(a*c)*A*c*d*abs(a)*abs(c)*e - A*a^2*c^2*e^2 - sqrt(a*c)*B*a*abs(a)*abs(c)*e^2)*arctan(sqrt(x*e + d)/sqrt(-(a
*c^2*d - sqrt(a^2*c^4*d^2 - (a*c^2*d^2 - a^2*c*e^2)*a*c^2))/(a*c^2)))/((a^2*c^2*e + sqrt(a*c)*a*c^2*d)*sqrt(-c
^2*d + sqrt(a*c)*c*e)*abs(a)) - 1/2*((x*e + d)^(3/2)*A*c*e - sqrt(x*e + d)*A*c*d*e + sqrt(x*e + d)*B*a*e^2)/((
(x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 - a*e^2)*a*c)

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maple [B]  time = 0.09, size = 432, normalized size = 1.92 \begin {gather*} \frac {A c d e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a}+\frac {A c d e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a}-\frac {B \,e^{2} \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {B \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\sqrt {e x +d}\, A d e}{2 \left (c \,e^{2} x^{2}-a \,e^{2}\right ) a}+\frac {A e \arctanh \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, a}-\frac {A e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, a}-\frac {\sqrt {e x +d}\, B \,e^{2}}{2 \left (c \,e^{2} x^{2}-a \,e^{2}\right ) c}-\frac {\left (e x +d \right )^{\frac {3}{2}} A e}{2 \left (c \,e^{2} x^{2}-a \,e^{2}\right ) a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+a)^2,x)

[Out]

-1/2*e/(c*e^2*x^2-a*e^2)*A/a*(e*x+d)^(3/2)+1/2*e/(c*e^2*x^2-a*e^2)/a*(e*x+d)^(1/2)*A*d-1/2*e^2/(c*e^2*x^2-a*e^
2)/c*(e*x+d)^(1/2)*B+1/2*e/a/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*
e^2)^(1/2))*c)^(1/2)*c)*A*c*d-1/4*e^2/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((
c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*B+1/4*e/a/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^
2)^(1/2))*c)^(1/2)*c)*A+1/2*e/a/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(
a*c*e^2)^(1/2))*c)^(1/2)*c)*A*c*d-1/4*e^2/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2
)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*B-1/4*e/a/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(
a*c*e^2)^(1/2))*c)^(1/2)*c)*A

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (B x + A\right )} \sqrt {e x + d}}{{\left (c x^{2} - a\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+a)^2,x, algorithm="maxima")

[Out]

integrate((B*x + A)*sqrt(e*x + d)/(c*x^2 - a)^2, x)

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mupad [B]  time = 4.10, size = 5062, normalized size = 22.50

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(1/2))/(a - c*x^2)^2,x)

[Out]

atan(((((64*B*a^4*c^2*e^4 - 64*A*a^3*c^3*d*e^3)/(8*a^3) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((4*A^2*a^3*c^5*d^3 +
B^2*a*e^3*(a^9*c^5)^(1/2) + A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*d*e^2 + B^2*a^5*c^3
*d*e^2 - 4*A*B*a^4*c^4*d^2*e - 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5*e^2)))^(1/2))*((4*A^2
*a^3*c^5*d^3 + B^2*a*e^3*(a^9*c^5)^(1/2) + A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*d*e^2
+ B^2*a^5*c^3*d*e^2 - 4*A*B*a^4*c^4*d^2*e - 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5*e^2)))^
(1/2) + ((d + e*x)^(1/2)*(4*A^2*c^3*d^2*e^2 + A^2*a*c^2*e^4 + B^2*a^2*c*e^4 - 4*A*B*a*c^2*d*e^3))/a^2)*((4*A^2
*a^3*c^5*d^3 + B^2*a*e^3*(a^9*c^5)^(1/2) + A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*d*e^2
+ B^2*a^5*c^3*d*e^2 - 4*A*B*a^4*c^4*d^2*e - 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5*e^2)))^
(1/2)*1i - (((64*B*a^4*c^2*e^4 - 64*A*a^3*c^3*d*e^3)/(8*a^3) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((4*A^2*a^3*c^5*
d^3 + B^2*a*e^3*(a^9*c^5)^(1/2) + A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*d*e^2 + B^2*a^
5*c^3*d*e^2 - 4*A*B*a^4*c^4*d^2*e - 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5*e^2)))^(1/2))*((
4*A^2*a^3*c^5*d^3 + B^2*a*e^3*(a^9*c^5)^(1/2) + A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*
d*e^2 + B^2*a^5*c^3*d*e^2 - 4*A*B*a^4*c^4*d^2*e - 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5*e^
2)))^(1/2) - ((d + e*x)^(1/2)*(4*A^2*c^3*d^2*e^2 + A^2*a*c^2*e^4 + B^2*a^2*c*e^4 - 4*A*B*a*c^2*d*e^3))/a^2)*((
4*A^2*a^3*c^5*d^3 + B^2*a*e^3*(a^9*c^5)^(1/2) + A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*
d*e^2 + B^2*a^5*c^3*d*e^2 - 4*A*B*a^4*c^4*d^2*e - 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5*e^
2)))^(1/2)*1i)/((4*A^3*c^2*d^2*e^3 - A^3*a*c*e^5 + A*B^2*a^2*e^5 - 4*A^2*B*a*c*d*e^4)/(4*a^3) + (((64*B*a^4*c^
2*e^4 - 64*A*a^3*c^3*d*e^3)/(8*a^3) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((4*A^2*a^3*c^5*d^3 + B^2*a*e^3*(a^9*c^5)
^(1/2) + A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*d*e^2 + B^2*a^5*c^3*d*e^2 - 4*A*B*a^4*c
^4*d^2*e - 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5*e^2)))^(1/2))*((4*A^2*a^3*c^5*d^3 + B^2*a
*e^3*(a^9*c^5)^(1/2) + A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*d*e^2 + B^2*a^5*c^3*d*e^2
- 4*A*B*a^4*c^4*d^2*e - 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5*e^2)))^(1/2) + ((d + e*x)^(
1/2)*(4*A^2*c^3*d^2*e^2 + A^2*a*c^2*e^4 + B^2*a^2*c*e^4 - 4*A*B*a*c^2*d*e^3))/a^2)*((4*A^2*a^3*c^5*d^3 + B^2*a
*e^3*(a^9*c^5)^(1/2) + A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*d*e^2 + B^2*a^5*c^3*d*e^2
- 4*A*B*a^4*c^4*d^2*e - 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5*e^2)))^(1/2) + (((64*B*a^4*
c^2*e^4 - 64*A*a^3*c^3*d*e^3)/(8*a^3) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((4*A^2*a^3*c^5*d^3 + B^2*a*e^3*(a^9*c^
5)^(1/2) + A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*d*e^2 + B^2*a^5*c^3*d*e^2 - 4*A*B*a^4
*c^4*d^2*e - 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5*e^2)))^(1/2))*((4*A^2*a^3*c^5*d^3 + B^2
*a*e^3*(a^9*c^5)^(1/2) + A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*d*e^2 + B^2*a^5*c^3*d*e
^2 - 4*A*B*a^4*c^4*d^2*e - 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5*e^2)))^(1/2) - ((d + e*x)
^(1/2)*(4*A^2*c^3*d^2*e^2 + A^2*a*c^2*e^4 + B^2*a^2*c*e^4 - 4*A*B*a*c^2*d*e^3))/a^2)*((4*A^2*a^3*c^5*d^3 + B^2
*a*e^3*(a^9*c^5)^(1/2) + A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*d*e^2 + B^2*a^5*c^3*d*e
^2 - 4*A*B*a^4*c^4*d^2*e - 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5*e^2)))^(1/2)))*((4*A^2*a^
3*c^5*d^3 + B^2*a*e^3*(a^9*c^5)^(1/2) + A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*d*e^2 +
B^2*a^5*c^3*d*e^2 - 4*A*B*a^4*c^4*d^2*e - 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5*e^2)))^(1/
2)*2i + atan(((((64*B*a^4*c^2*e^4 - 64*A*a^3*c^3*d*e^3)/(8*a^3) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((4*A^2*a^3*c
^5*d^3 - B^2*a*e^3*(a^9*c^5)^(1/2) - A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*d*e^2 + B^2
*a^5*c^3*d*e^2 - 4*A*B*a^4*c^4*d^2*e + 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5*e^2)))^(1/2))
*((4*A^2*a^3*c^5*d^3 - B^2*a*e^3*(a^9*c^5)^(1/2) - A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c
^4*d*e^2 + B^2*a^5*c^3*d*e^2 - 4*A*B*a^4*c^4*d^2*e + 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5
*e^2)))^(1/2) + ((d + e*x)^(1/2)*(4*A^2*c^3*d^2*e^2 + A^2*a*c^2*e^4 + B^2*a^2*c*e^4 - 4*A*B*a*c^2*d*e^3))/a^2)
*((4*A^2*a^3*c^5*d^3 - B^2*a*e^3*(a^9*c^5)^(1/2) - A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c
^4*d*e^2 + B^2*a^5*c^3*d*e^2 - 4*A*B*a^4*c^4*d^2*e + 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5
*e^2)))^(1/2)*1i - (((64*B*a^4*c^2*e^4 - 64*A*a^3*c^3*d*e^3)/(8*a^3) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((4*A^2*
a^3*c^5*d^3 - B^2*a*e^3*(a^9*c^5)^(1/2) - A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*d*e^2
+ B^2*a^5*c^3*d*e^2 - 4*A*B*a^4*c^4*d^2*e + 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5*e^2)))^(
1/2))*((4*A^2*a^3*c^5*d^3 - B^2*a*e^3*(a^9*c^5)^(1/2) - A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*
a^4*c^4*d*e^2 + B^2*a^5*c^3*d*e^2 - 4*A*B*a^4*c^4*d^2*e + 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^
7*c^5*e^2)))^(1/2) - ((d + e*x)^(1/2)*(4*A^2*c^3*d^2*e^2 + A^2*a*c^2*e^4 + B^2*a^2*c*e^4 - 4*A*B*a*c^2*d*e^3))
/a^2)*((4*A^2*a^3*c^5*d^3 - B^2*a*e^3*(a^9*c^5)^(1/2) - A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*
a^4*c^4*d*e^2 + B^2*a^5*c^3*d*e^2 - 4*A*B*a^4*c^4*d^2*e + 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^
7*c^5*e^2)))^(1/2)*1i)/((4*A^3*c^2*d^2*e^3 - A^3*a*c*e^5 + A*B^2*a^2*e^5 - 4*A^2*B*a*c*d*e^4)/(4*a^3) + (((64*
B*a^4*c^2*e^4 - 64*A*a^3*c^3*d*e^3)/(8*a^3) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((4*A^2*a^3*c^5*d^3 - B^2*a*e^3*(
a^9*c^5)^(1/2) - A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*d*e^2 + B^2*a^5*c^3*d*e^2 - 4*A
*B*a^4*c^4*d^2*e + 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5*e^2)))^(1/2))*((4*A^2*a^3*c^5*d^3
- B^2*a*e^3*(a^9*c^5)^(1/2) - A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*d*e^2 + B^2*a^5*c
^3*d*e^2 - 4*A*B*a^4*c^4*d^2*e + 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5*e^2)))^(1/2) + ((d
+ e*x)^(1/2)*(4*A^2*c^3*d^2*e^2 + A^2*a*c^2*e^4 + B^2*a^2*c*e^4 - 4*A*B*a*c^2*d*e^3))/a^2)*((4*A^2*a^3*c^5*d^3
- B^2*a*e^3*(a^9*c^5)^(1/2) - A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*d*e^2 + B^2*a^5*c
^3*d*e^2 - 4*A*B*a^4*c^4*d^2*e + 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5*e^2)))^(1/2) + (((6
4*B*a^4*c^2*e^4 - 64*A*a^3*c^3*d*e^3)/(8*a^3) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((4*A^2*a^3*c^5*d^3 - B^2*a*e^3
*(a^9*c^5)^(1/2) - A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*d*e^2 + B^2*a^5*c^3*d*e^2 - 4
*A*B*a^4*c^4*d^2*e + 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5*e^2)))^(1/2))*((4*A^2*a^3*c^5*d
^3 - B^2*a*e^3*(a^9*c^5)^(1/2) - A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*d*e^2 + B^2*a^5
*c^3*d*e^2 - 4*A*B*a^4*c^4*d^2*e + 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5*e^2)))^(1/2) - ((
d + e*x)^(1/2)*(4*A^2*c^3*d^2*e^2 + A^2*a*c^2*e^4 + B^2*a^2*c*e^4 - 4*A*B*a*c^2*d*e^3))/a^2)*((4*A^2*a^3*c^5*d
^3 - B^2*a*e^3*(a^9*c^5)^(1/2) - A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*d*e^2 + B^2*a^5
*c^3*d*e^2 - 4*A*B*a^4*c^4*d^2*e + 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5*e^2)))^(1/2)))*((
4*A^2*a^3*c^5*d^3 - B^2*a*e^3*(a^9*c^5)^(1/2) - A^2*c*e^3*(a^9*c^5)^(1/2) + 2*A*B*a^5*c^3*e^3 - 3*A^2*a^4*c^4*
d*e^2 + B^2*a^5*c^3*d*e^2 - 4*A*B*a^4*c^4*d^2*e + 2*A*B*c*d*e^2*(a^9*c^5)^(1/2))/(64*(a^6*c^6*d^2 - a^7*c^5*e^
2)))^(1/2)*2i - ((A*e*(d + e*x)^(3/2))/(2*a) + ((B*a*e^2 - A*c*d*e)*(d + e*x)^(1/2))/(2*a*c))/(c*(d + e*x)^2 -
a*e^2 + c*d^2 - 2*c*d*(d + e*x))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(1/2)/(-c*x**2+a)**2,x)

[Out]

Timed out

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